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Chapter 5: Problem 33

Find each product. $$ (4 m+3)\left(5 m^{3}-4 m^{2}+m-5\right) $$

Short Answer

Expert verified
20m^4 - m^3 - 8m^2 - 17m - 15

Step by step solution

01

Distribute the first term

Multiply each term inside the second polynomial \(5 m^{3}-4 m^{2}+m-5\) by the first term \(4m\) from the first polynomial. \[ (4m) \times (5m^3) = 20m^4 \ (4m) \times (-4m^2) = -16m^3 \ (4m) \times (m) = 4m^2 \ (4m) \times (-5) = -20m \]
02

Distribute the second term

Multiply each term inside the polynomial \(5 m^{3}-4 m^{2}+m-5\) by the second term \(3\) from the first polynomial. \[ (3) \times (5m^3) = 15m^3 \ (3) \times (-4m^2) = -12m^2 \ (3) \times (m) = 3m \ (3) \times (-5) = -15 \]
03

Combine like terms

Write down all the distributed terms and then combine the like terms. \[ 20m^4 - 16m^3 + 4m^2 - 20m + 15m^3 - 12m^2 + 3m - 15 \ Combining like terms: \ 20m^4 + (-16m^3 + 15m^3) + (4m^2 - 12m^2) + (-20m + 3m) - 15 \ Which simplifies to: 20m^4 - m^3 - 8m^2 - 17m - 15 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

distributive property
The distributive property is a key concept in algebra. It allows you to multiply a single term by each term within a parenthesis to simplify expressions. This is crucial when dealing with polynomial multiplication.

Here鈥檚 how it works with our example:
  1. First, distribute the first term in the outer polynomial
    • For instance, distribute the term 4m across each term in (5m^3 - 4m^2 + m - 5)
  2. Next, repeat this step with the second term in the outer polynomial
    • Distribute the constant 3 across (5m^3 - 4m^2 + m - 5)
Each distribution step results in a new term. Remember to respect the signs (positive/negative) of each term when you distribute.
combining like terms
Once you've distributed all terms, you鈥檒l get a list of terms. Combining like terms is your next step.

'Like terms' are terms that have the same variables raised to the same powers. For example, 3m^3 and -4m^3 are like terms, but 3m^3 and 4m^2 are not.
  • Identify all like terms, e.g., 15m^3 and -16m^3
  • Add or subtract the coefficients (numerical factors) of these like terms.
Combining like terms simplifies the polynomial, making it easier to work with and understand.
algebraic expressions
Algebraic expressions are combinations of variables, numbers, and operations (+, -, *, /). Multiplying two polynomials generates a more complex algebraic expression.

For our given example, multiplying (4m + 3) by (5m^3 - 4m^2 + m - 5) produced terms like 20m^4 and -17m.
  • Each term consists of a coefficient (numerical part) and a variable part.
  • Some terms have variables raised to different powers.
This expression can be simplified by distributing and then combining like terms to reduce it to a simpler state while maintaining its value.

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Most popular questions from this chapter

Evaluate. $$ 277 \div 1000 $$

Perform each division. $$ \left(3 a^{2}-11 a+17\right) \div(2 a+6) $$

Multiply. $$ -3\left(x^{2}+7\right) $$

Use scientific notation to calculate the answer to each problem. In \(2008,\) the U.S. government collected about \(\$ 4013\) per person in personal income taxes. If the population was \(304,000,000,\) how much did the government collect in taxes for \(2008 ?\)

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ $$ \text { can be used to perform some multiplication problems. Here are two examples.} $$ $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1^{2} \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 20 \frac{1}{2} \times 19 \frac{1}{2} $$
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